The trissotetras: or, a most exquisite table for resolving all manner of triangles, whether plaine or sphericall, rectangular or obliquangular, with greater facility, then ever hitherto hath been practised: most necessary for all such as would attaine to the exact knowledge of fortification, dyaling, navigation, surveying, architecture, the art of shadowing, taking of heights, and distances, the use of both the globes, perspective, the skill of making the maps, the theory of the planets, the calculating of their motions, and of all other astronomicall computations whatsoever. Now lately invented, and perfected, explained, commented on, and with all possible brevity, and perspicuity, in the hiddest, and most re-searched mysteries, from the very first grounds of the science it selfe, proved, and convincingly demonstrated. / By Sir Thomas Urquhart of Cromartie Knight. Published for the benefit of those that are mathematically affected.

The Loxogonosphericall Triangles, whether Amblygonosphericall or Oxygono∣sphericall, are either Monurgetick or Disergetick.

THe Monurgetick have two figures, Datamista and Datapu∣ra.

Datamista is of all those Loxogonospherical Monurgetick pro∣blems, wherein the Angles and sides being intermixedly given, (and therefore one of them being alwaies of another kind from the other two) either an Angle, or a side is demanded: it hath two Moods, Lamaneprep, and Menerolo.

The first Mood Lamaneprep, comprehendeth all those Loxogo∣nosphericall problems, wherein two angles being given, and an opposit side, another opposit side is demanded, and by its Resolver, Sapeg—Se—Sapy—☞Syr, sheweth, that if to the Logarithms of the sine of the side given, and sine of the Angle opposit, to the side required, we joyne the Arithmeticall complement of the sine of the Angle opposit, to the proposed side (which is the refined Anti∣secant) we will thereby attain to the knowledge of the sine of the side demanded. The reason of this is grounded on the third A∣xiom, Seprosa, as you may perceive by the first syllable of the Ob∣liquangularie directory, Lame.

The second Mood of this figure is Menerolo, wich comprehend∣eth all those Amblygonosphericall problems, wherein two sides being given with an opposit angle, another opposit angle is de∣manded, and by its Resolver Sepag—Sa—Sepi☞Sir, sheweth, that if to the summe of the Logarithms of the sine of the given angle, and sine of the side opposit to the angle required, we joyne the Arithmeticall complement of the sine of the side opposit to the given angle (which is the refined Cosecant of the said angle) it will afford us the sine of the angle required. The reason of this operation is grounded on the third Axiom of Sphericalls, a pro∣gresse in sines shewing clearely, how that both this, and the for∣mer, doe totally depend on the Axiom of Seprosa, as is evident by the second syllable of its directorie, Lame.

The second figure of the Monurgetick Loxogonosphericalls

treateth of all those questions, wherein the Datas being either sides alone, or Angles alone, an Angle or a side is demanded. This Figure of Datapura is divided into two Moods, viz. Nerelema, and Ralamane, which are of such affinity, that upon one and the same Theorem dependeth the Analogy that resolveth both.

The first Mood thereof, Nerelema, comprehendeth all those Problems, wherein the three sides being given, an Angle is deman∣ded, and is the third of the Monurgeticks, as by its Characte∣ristick the third Liquid is perceivable.

The curteous Reader may be pleased to take notice, that in both the Moods of the Datapurall Figure, I am in some measure neces∣sitated for the better order sake, to couch two precepts, or docu∣ments, for the Faciendas thereof, and to premise that one concer∣ning the three Legs given, before I make any mention of the maine Resolver, whereupon both the foresaid Moods are founded, to which Resolver, because of both their dependences on it, I have allowed here in the Glosse, the same middle place, which it pos∣sesseth in the Table of my Trissotetras.

The precept of Nerelema is Halbasalzes * Ad* Ab* Sadsabrere∣galsbis Ir: that is to say, for the finding out of an Angle when the three Legs are given, as soone as we have constituted the sustenta∣tive Leg of that Angle a Base, the halfe thereof must be taken, and to that halfe we must adde halfe the difference of the other two Legs, and likewise from that halfe subtract the half difference of the fore∣said two Legs, then the summe and the residue being two Arches, we must, to the Logarithms of the Sine of the summe, and Sine of the Remainer, joyne the Logarithms of the Arithmeticall complements of the Sines of the sides, which are the refined An∣tisecants of the said Legs, and halfe that summe will afford, us the Logarithm of the Sine of an Arch, which doubled, is the verticall Angle, we demand; for out of its Resolver, Parses—Powto—Par∣sadsab☞Powsalvertir, is the Analogy of the former work made cleare, the Theorem being, As the Oblong or Parallelogram contained under the Sines of the Legs, to the square, power or quadrat of the totall Sine: so the Rectangle, or Oblong made of the right Sines of the sum, and difference of the halfe Base, and difference of the Legs, to the square of the right Sine of halfe the verticall Angle.

The reason hereof will be manifest enough to the industrious

Reader, if when by a peculiar Diagram, of whose equiangular Tri∣angles the foresaid Sines and differences are made the constitutive sides, he hath evinced their Analogy to one another, he be then pleased to perpend, how, in two rowes of proportionall numbers, the products arising of the homologall roots, are in the same pro∣portion amongst themselves, that the said roots towards one ano∣ther are; wherewithall if he doe consider, how the halfs must needs keep the same proportion that their wholes; and then, in the work it selfe of collationing severall orders of proportionall termes, both single and compound, be carefull to dash out a divider against a multiplyer, and afterwards proceed in all the rest, according to the ordinary rules of Aequation, and Analogy, he cannot choose but extricat himselfe with ease forth of all the windings of this elaboured proposition.

Upon this Theorem (as I have told you) dependeth likewise the Document for the faciendum of Ralamane, which is the second Mood of Datapura, and the last of the Monurgetick Loxogono∣sphericals, as is pointed at by Nera the Directory therof. This Mood Ralamane comprehendeth all those Loxogonosphericall Problems, wherein the three Angles being given, a side is de∣manded. And by its Resolver, Parses—Powto—Parsadsab☞Powsalvertir, according to the peculiar precept of this Mood Kourbfasines (Ereled) Koufbraxypopyx, sheweth, that if we take the complement to a semicircle of the Angle opposite to the side re∣quired, which for distinction sake we doe here call the Base; and frame, of the foresaid complement to a semicircle, a second Base for the fabrick of a new Triangle, whose other two sides have the graduall measure of the former Triangles other two Angles: (and so the three Angles being converted into sides) the com∣plement to a Semicircle of the new Verticall, or Angle opposite to the new Base, will be the measure of the true Base or Leg requi∣red, and the Angle insident on the right end of the new Base in the second Triangle, falleth to be the side conterminall with the left end of the true Base in the first Triangle, and the Angle ad∣joyning the left end of the false Base in the second Triangle, be∣comes the side adjacent to the right end of the old Base in the first Triangle. So that thus by the Angles all andeach of the sides are found out, all which works are to be performed by the preceding

Mood, upon the Theorem, whereof the reason of all these opera∣tions doth depend.